On the uniformity of distribution of the RSA pairs

Let $m= pl$ be a product of two distinct primes $p$ and $l$. We show that for almost all exponents $e$ with $\operatorname{gcd} (e, \varphi(m))= 1$the RSA pairs $(x, x^e)$ are uniformly distributed modulo $m$when $x$ runs through

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the group of units $\mathbb{Z}_m^*$ modulo $m$ (that is, as in the classical RSA scheme);

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the set of $k$-products $x = a_{i_1}\cdots a_{i_k}$, $1 \le i_1 < \cdots < i_k \le n$, where $a_1, \cdots, a_n \in \mathbb{Z}_m^*$ are selected at random (that is, as in the recently introduced RSA scheme with precomputation).

These results are based on some new bounds of exponential sums.